Preview Activity 3.2.1.
In Desmos, define \(g(t) = ab^t+c\) and accept the prompt for sliders for \(a\text{,}\) \(b\text{,}\) and \(c\text{.}\) Edit the sliders so that \(a\) has values from \(a = 5\) to \(a = 50\text{,}\) \(b\) has values from \(b = 0.7\) to \(b = 1.3\text{,}\) and \(c\) has values from \(c = -5\) to \(c = 5\) (each with a step-size of 0.01). In addition, in Desmos let \(P = (0, g(0))\) and check the box to show the label. Finally, zoom out so that the window shows an interval of \(t\)-values from \(-30 \le t \le 30\text{.}\)
(a)
Set \(b = 1.1\) and explore the effects of changing the values of \(a\) and \(c\text{.}\) Write several sentences to summarize your observations.
(b)
Follow the directions for (a) again, this time with \(b = 0.9\)
(c)
Set \(a = 5\) and \(c = 4\text{.}\) Explore the effects of changing the value of \(b\text{;}\) be sure to include values of \(b\) both less than and greater than 1. Write several sentences to summarize your observations.
(d)
When \(0 \lt b \lt 1\text{,}\) what happens to the graph of \(g\) when we consider positive \(t\)-values that get larger and larger?
